# Modus Tollendo Ponens/Variant/Formulation 1

## Theorem

$p \lor q \dashv \vdash \neg p \implies q$

This can be expressed as two separate theorems:

### Forward Implication

$p \lor q \vdash \neg p \implies q$

### Reverse Implication

$\neg p \implies q \vdash p \lor q$

Note that the latter proof requires Law of Excluded Middle.

Therefore it is not valid in intuitionistic logic.

## Proof

We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.

$\begin{array}{|ccc||cccc|} \hline p & \lor & q & \neg & p & \implies & q \\ \hline F & F & F & T & F & F & F \\ F & T & T & T & F & T & T \\ T & T & F & F & T & T & F \\ T & T & T & F & T & T & T \\ \hline \end{array}$

$\blacksquare$